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Some New Hilbert's Type Inequalities
Journal of Inequalities and Applications volume 2009, Article number: 851360 (2009)
Abstract
Some new inequalities similar to Hilbert's type inequality involving series of nonnegative terms are established.
1. Introduction
In recent years, several authors [1–10] have given considerable attention to Hilbert's type inequalities and their various generalizations. In particular, in [1], Pachpatte proved somenew inequalities similar to Hilbert's inequality [11, page 226] involving series of nonnegative terms. The main purpose of this paper is to establish their general forms.
2. Main Results
In [1], Pachpatte established the following inequality involving series of nonnegative terms.
Theorem 2 A.
Let and let
and
be two nonnegative sequences of real numbers defined for
and
, where
are natural numbers. Let
and
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851360/MediaObjects/13660_2008_Article_2017_Equ1_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851360/MediaObjects/13660_2008_Article_2017_Equ2_HTML.gif)
We first establish the following general form of inequality (2.1).
Theorem 2.1.
Let and
. Let
and
be positive sequences of real numbers defined for
and
, where
are natural numbers. Let
and
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851360/MediaObjects/13660_2008_Article_2017_Equ3_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851360/MediaObjects/13660_2008_Article_2017_Equ4_HTML.gif)
Proof.
By using the following inequality (see [12]):
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851360/MediaObjects/13660_2008_Article_2017_Equ5_HTML.gif)
where is a constant, and
,
, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851360/MediaObjects/13660_2008_Article_2017_Equ6_HTML.gif)
Similarly, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851360/MediaObjects/13660_2008_Article_2017_Equ7_HTML.gif)
From (2.6) and (2.7), using Hölder's inequality [13] and the elementary inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851360/MediaObjects/13660_2008_Article_2017_Equ8_HTML.gif)
where , and
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851360/MediaObjects/13660_2008_Article_2017_Equ9_HTML.gif)
Dividing both sides of (2.9) by summing up over
from 1 to
first, then summing up over
from 1 to
, using again Hölder's inequality, then interchanging the order of summation, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851360/MediaObjects/13660_2008_Article_2017_Equ10_HTML.gif)
This completes the proof.
Remark 2.2.
Taking , (2.3) becomes
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851360/MediaObjects/13660_2008_Article_2017_Equ11_HTML.gif)
Taking and changing
,
,
and
into
,
,
and
respectively, and with suitable changes, (2.11) reduces to Pachpatte [1, inequality (1)].
In [1], Pachpatte also established the following inequality involving series of nonnegative terms.
Theorem 2 B.
Let ,
be as defined in Theorem A. Let
and
be positive sequences for
and
where
are natural numbers. Define
and
. Let
and
be real-valued, nonnegative, convex, submultiplicative functions defined on
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851360/MediaObjects/13660_2008_Article_2017_Equ12_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851360/MediaObjects/13660_2008_Article_2017_Equ13_HTML.gif)
Inequality (2.12) can also be generalized to the following general form.
Theorem 2.3.
Let ,
,
and
be as defined in Theorem 2.1. Let
and
be positive sequences for
and
. Define
and
Let
and
be real-valued, nonnegative, convex, submultiplicative functions defined on
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851360/MediaObjects/13660_2008_Article_2017_Equ14_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851360/MediaObjects/13660_2008_Article_2017_Equ15_HTML.gif)
Proof.
By the hypotheses, Jensen's inequality, and Hölder's inequality, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851360/MediaObjects/13660_2008_Article_2017_Equ16_HTML.gif)
Similarly,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851360/MediaObjects/13660_2008_Article_2017_Equ17_HTML.gif)
By (2.16) and (2.17), and using the elementary inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851360/MediaObjects/13660_2008_Article_2017_Equ18_HTML.gif)
where and
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851360/MediaObjects/13660_2008_Article_2017_Equ19_HTML.gif)
Dividing both sides of (2.19) by and summing up over
from 1 to
first, then summing up over
from 1 to
, using again inverse Hölder's inequality, and then interchanging the order of summation, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851360/MediaObjects/13660_2008_Article_2017_Equ20_HTML.gif)
The proof is complete.
Remark 2.4.
Taking , (2.14) becomes
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851360/MediaObjects/13660_2008_Article_2017_Equ21_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851360/MediaObjects/13660_2008_Article_2017_Equ22_HTML.gif)
Taking and changing
,
,
and
into
,
,
and
respectively, and with suitable changes, (2.21) reduces to Pachpatte [1, Inequality ( 7)].
Theorem 2.5.
Let ,
,
,
,
and
,
be as defined in Theorem 2.3. Define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851360/MediaObjects/13660_2008_Article_2017_Equ23_HTML.gif)
for and
where
are natural numbers. Let
and
be real-valued, nonnegative, convex functions defined on
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851360/MediaObjects/13660_2008_Article_2017_Equ24_HTML.gif)
Proof.
By the hypotheses, Jensen's inequality, and Hölder's inequality, it is easy to observe that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851360/MediaObjects/13660_2008_Article_2017_Equ25_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851360/MediaObjects/13660_2008_Article_2017_Equ26_HTML.gif)
Proceeding now much as in the proof of Theorems 2.1 and 2.3, and with suitable modifications, it is not hard to arrive at the desired inequality. The details are omitted here.
Remark 2.6.
In the special case where and
, Theorem 2.5 reduces to the following result.
Theorem 2 C.
Let be as defined in Theorem B. Define
and
for
and
, where
are natural numbers. Let
and
be real-valued, nonnegative, convex functions defined on
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851360/MediaObjects/13660_2008_Article_2017_Equ27_HTML.gif)
This is the new inequality of Pachpatte in [1,Theorem 4].
Remark 2.7.
Taking and
in Theorem 2.5, and in view of
, we obtain the following theorem.
Theorem 2 D.
Let ,
be as defined in Theorem A. Define
and
for
and
, where
are natural numbers. Let
and
be real-valued, nonnegative, convex functions defined on
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851360/MediaObjects/13660_2008_Article_2017_Equ28_HTML.gif)
This is the new inequality of Pachpatte in [1,Theorem 3].
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Acknowledgments
Research is supported by Zhejiang Provincial Natural Science Foundation of China(Y605065), Foundation of the Education Department of Zhejiang Province of China (20050392). Research is partially supported by the Research Grants Council of the Hong Kong SAR, China (Project no. HKU7016/07P) and a HKU Seed Grant forBasic Research.
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Zhao, CJ., Cheung, WS. Some New Hilbert's Type Inequalities. J Inequal Appl 2009, 851360 (2009). https://doi.org/10.1155/2009/851360
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DOI: https://doi.org/10.1155/2009/851360